Is there a way to determine the left 3D view sidebar's current width with python?
You can query the region's width from bpy.types.Region.width

I'm not sure if the toolbar region is always the second region (region[1]) in the bpy.types.Area.regions list, but you can always just loop through and check the type:

for region in bpy.context.area.regions:
    if region.type == "TOOLS":
        width = region.width
        break


That'll pick the first region of type TOOLS, though afaik there can be only one.